Mathematical Economics dr Wioletta Nowak Lecture 2
The Utility Function, Examples of Utility Functions: Normal Good, Perfect Substitutes, Perfect Complements, The Quasilinear and Homothetic Utility Functions, The Marginal Utility and The Marginal Rate of Substitution, The Optimal Choice, The Utility Maximization Problem, The Lagrange Method
The Utility Function A utility is a measure of the relative satisfaction from consumption of various goods. A utility function is a way of assigning a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers then less-preferred bundles.
The Utility Function The numerical magnitudes of utility levels have no intrinsic meaning the only property of a utility assignment that is important is how it orders the bundles of goods. The magnitude of the utility function is only important insofar as it ranks the different consumption bundles. Ordinal utility - consumer assigns a higher utility to the chosen bundle than to the rejected. Ordinal utility captures only ranking and not strength of preferences. Cardinal utility theories attach a significance to the magnitude of utility. The size of the utility difference between two bundles of goods is supposed to have some sort of significance.
Existence of a Utility Function Suppose preferences are complete, reflexive, transitive, continuous, and strongly monotonic. Then there exists a continuous utility function u : 2 which represents those preferences.
The Utility Function A utility function is a function u assigning a real number to each consumption bundle so that for a pair of bundles x and y:
Examples of Utility Functions
The Quasilinear Utility Function The quasilinear (partly linear) utility function is linear in one argument. For example the utility function linear in good 2 is the following: u 2 1 2 x1, x v x x
The Quasilinear Utility Function Specific examples of quasilinear utility would be: or u 2 1 2 x1, x x x u x, x ln x x 1 2 1 2
The Homothetic Utility Function
The Homothetic Utility Function
The Homothetic Utility Function Slopes of indifference curves are constant along a ray through the origin. Assuming that preferences can be represented by a homothetic function is equivalent to assuming that they can be represented by a function that is homogenous of degree 1 because a utility function is unique up to a positive monotonic transformation.
The Marginal Utility
The Marginal Rate of Substitution Suppose that we increase the amount of good i; how does the consumer have to change their consumption of good j in order to keep utility constant?
The Marginal Rate of Substitution
The Optimal Choice Consumers choose the most preferred bundle from their budget sets. The optimal choice of consumer is that bundle in the consumer s budget set that lies on the highest indifference curve.
The Optimal Choice
The Optimal Choice
The Optimal Choice Utility functions Budget line
The Optimal Choice
The Utility Maximization The problem of utility maximization can be written as: Consumers seek to maximize utility subject to their budget constraint. The consumption levels which solve the utility maximization problem are the Marshallian demand functions.
The Lagrange Method The method starts by defining an auxiliary function known as the Lagrangean: The new variable l is called a Lagrange multiplier since it is multiplied by constraint.
The Lagrange Method